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We replace the discrete optimization (REF ) by an annealed sequence of differentiable inequality constraints that converges to (REF ) when the annealing parameter goes to 0.

We use a novel first-order algorithm proposed in [1]} to solve the relaxed subproblems in the annealed sequence, leading to ASkewSGD. Unlike classical constrained optimization algorithms, including the projection method or sequential quadratic programming [2]}, this approach relies exclusively on local approximations of the feasible set. This local approximation includes only the active constraints, and is guaranteed to be a convex polyhedron even if the underlying constraint set is non convex. This makes the resulting algorithm easy to implement and also ensures that the descent is not stopped as soon as a new constraint is violated.

We show how ASkewSGD can be applied to train QNN. The complexity of the resulting algorithm is similar to that of BC or LUQ [3]}, [4]} and ProxQuant [5]}. Our algorithm uses high precision latent weights and uses classical backpropagation to evaluate the gradients.

We provide convergence guarantees for ASkewSGD. We stress that, as opposed to [1]}, no convexity assumption on the objective function or the feasible set is made.

We evaluate the performance of ASkewSGD on classical computer vision datasets using ConvNets and ResNets. Our experiments show that QNNs trained with ASkewSGD achieve accuracy very close to that of their floating-point counterparts, and outperform or are on par with comparable baselines.

